Rotating Electrical Machines E-Book

René Le Doeuff

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 208

Veröffentlichungsjahr: 2013

Beschreibung

In this book a general matrix-based approach to modeling electrical machines is promulgated. The model uses instantaneous quantities for key variables and enables the user to easily take into account associations between rotating machines and static converters (such as in variable speed drives). General equations of electromechanical energy conversion are established early in the treatment of the topic and then applied to synchronous, induction and DC machines. The primary characteristics of these machines are established for steady state behavior as well as for variable speed scenarios. Important new applications for this technology (such as wind turbines, electric propulsion systems for large ships, etc.) are addressed and the book is illustrated with a large number of informative and detailed photographs, provided by various companies at the leading edge of research and applications in the field.

Leseprobe

Table of Contents

Preface

Chapter 1. Main Requirements

1.1. Introduction

1.2. Sinusoidal variables

1.3. Electromagnetism

1.4. Power electronics

Chapter 2. Introduction to Rotating Electrical Machines

2.1. Introduction

2.2. Main notations

2.3. Principle of the electromechanical energy conversion

2.4. Continuous energy conversion

2.5. Non-salient and salient poles

2.6. Notion of pole pitch

2.7. Stator/rotor coupling: the “basic machine”

2.8. Losses within the machines

2.9. Nominal values

2.10. General sign covenant

2.11. Establishment of matricial equations

2.12. Mechanical equation

2.13. Conclusion

Chapter 3. Synchronous Machines

3.1. Introduction

3.2. Introduction and equations of the cylindrical synchronous machine

3.3. Analysis of the synchronous machine connected to an infinite power network

3.4. Considerations about the salient pole synchronous machine

3.5. Consideration about permanent magnet machines

3.6. Inverted AC generators

3.7. Implementation of synchronous machines

3.8. Experimental determination of the parameters

Chapter 4. Induction Machines

4.1. Introduction

4.2. General considerations

4.3. Equations

4.4. Equivalent circuits

4.5. Induction machine torque

4.6. Study of the stability

4.7. Circle diagram (or “Blondel” diagram)

4.8. Induction machine characteristics

4.9. Implementation of induction machines

4.10. Principle of the experimental determination of the parameters

Chapter 5. Direct Current Machines

5.1. Introduction

5.2. Main notations

5.3. DC machine structure

5.4. DC machine equations

5.5. Separately excited motor

5.6. Series excited motor

5.7. Special case of the series motor: the universal motor

5.8. Commutation phenomena

5.9. Saturation and armature reaction

First published 2010 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.

Adapted and updated from Machines électriques tournantes : de la modélisation matricielle à la mise en œuvre published 2009 in France by Hermes Science/Lavoisier © LAVOISIER 2009

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:

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© ISTE Ltd 2010

The rights of René Le Doeuff and Mohamed El Hadi Zaïm to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988.

Library of Congress Cataloging-in-Publication Data

Le Doeuff, R.

 [Machines electriques tournantes. English]

 Rotating electrical machines / René Le Doeuff, Mohamed El Hadi Zaïm.

   p. cm.

 Includes bibliographical references and index.

 ISBN 978-1-84821-169-8

 1. Electric motors 2. Electric machinery--Design and construction. 3. Industrial equipment--Design and construction. I. El Hadi Zaom, Mohamed. II. Title.

 TK2435.D6413 2009

 621.46--dc22

2009039568

British Library Cataloguing-in-Publication Data

A CIP record for this book is available from the British Library

ISBN 978-1-84821-169-8

Preface

Rotating electrical machines provide the basis of the electromechanical energy conversion and constitute the core of a wide scientific and technological field called “electrical engineering”. This discipline has seen a very important evolution with the extensive development of related fields: power electronics, analogical and digital control techniques, etc. This revolution has led to the generalization of electrical actuators in every industrial area as well as in everyday life. It has also modified the way the machines are used while, at the same time, simplifying their adaptation to new energy sources. Therefore, this evolution has to be taken into account in the teaching of electrical machinery.

The present text is the result of our long teaching and research experience in various universities' engineering schools, both in France and Algeria. It is intended mainly for Master's level students enrolled in electrical engineering programs. Its aims consists of providing readers with the essential knowledge of electrical machines, their structures, the ways they can be modeled and their implementation. This basic understanding should allow them to tackle with relative ease the study of transient phenomenon, speed variation and control of drives, and any other special applications.

This methodological approach was first proposed by Professor E. J. Gudefin in Nancy (France) in the 1960s. It is based on matrix representation of the machine equations using instantaneous values of electromagnetic variables.

This modeling approach is particularly suitable for the study of electrical machines fed by static converters; and it is necessary for the analysis of machines in transient regimes or any other study that uses Concordia and Park transformations, etc. It can also be used to establish classic steady state equations of electrical motors. The calculation of the instantaneous electromagnetic torque leads to a simple and convenient representation of the association machine-converter enabling an easy understanding of the continuous energy conversion phenomenon.

The main preliminary knowledge useful for reading this text (electromagnetism, sinusoidal systems, power electronics) is gathered in Chapter 1 (Main Requirements).

General concepts are established in Chapter 2 (Introduction to Rotating Electrical Machines) and are then used for different analyses of conventional machines: Synchronous Machines (Chapter 3), Induction Machines (Chapter 4) and Direct Current Machines (Chapter 5). Many examples describing the use of these machines with and without converters are also presented. Some traditional aspects (e.g. resistive starters, circle diagrams, etc.), which are of very little use today, are still presented because of their historical and pedagogical interest.

To make this book as factual as possible, we have illustrated it with many photographs that have been graciously provided by industrial firms; most of the curves, diagrams and characteristics are those of machines that really exist. Different field distribution plots describing electromagnetic behaviours of machines have been obtained from software codes developed in our research laboratory.

We would like to acknowledge all the individuals and organizations who took part in the realization of this book:

– Our colleagues from Electrical Engineering Department of Polytech'Nantes (France), particularly Professors M.F. Benkhoris and M. Machmoum.

– Professor Bernard Multon, from École Normale Supérieure de Cachan (France).

– Professor Guy Olivier from École Polytechnique de Montréal (Canada).

– The following firms: ECA EN, Converteam and STX France (previously, Aker Yards) for generously providing most of the photographs illustrating this book.

The authors wish to pay a particular tribute to their mentor, the late Professor Emeritus Edmond J. Gudefin (1923-1996).

Chapter 1

Main Requirements

1.1. Introduction

The study of rotating electrical machines is a science which is linked with several other topics. In order to make this book easier to read, we are going to summarize the main results and concepts used later on in this introductory chapter:

– sinusoidal systems;

– electromagnetism;

– power electronics.

1.2. Sinusoidal variables

1.2.1. Single-phase variables

1.2.1.1. Timed expressions

An x variable, a timed-sinusoidal function, can be written as:

where X is the rms (root-mean-square) value and is the angular velocity.

1.2.1.2. Vector representation

The x variable defined above can be considered to be the projection on an axis of a vector of length X rotating anticlockwise at an angular velocity ω (Figure 1.1).

Figure 1.1.Vector representation of a sinusoidal variable

1.2.1.3. Single-phase currents and voltages

If a sinusoidal single-phase voltage v is applied at a Z impedance terminal, current i in this impedance, at steady state, is also sinusoidal, and can be written:

φ being the phase shift between the voltage often chosen as the origin and the current. Conventionally, φ is counted positively when the current is lagging behind the voltage. The instantaneous power supplied to impedance Z is:

[1.1]

is the active power and:

[1.2]

is the pulsating power. It must be noted that this variable, which characterizes the fact that the single-phase power supplied to a receiver is time varying, is cancelled with balanced polyphase systems.

Figure 1.2 shows that voltage is changed into current through a similitude of ratio Z and angle φ.

Figure 1.2.Vector representation of sinusoidal current and voltage

1.2.1.4. Complex representation

Complex numbers are very useful to represent the previous similitude and vector will thus be associated with complex number as well as complex number Ī with vector . They can then be written as follows:

Complex impedance is also defined by ratio:

It will be set down:

R and X respectively being the resistance and the reactance expressed in Ohms.

is also introduced :

[1.3]

is the apparent power expressed in volt-amperes (VA). Q is the reactive power expressed in volt-amperes reactives (VAr).

1.2.2. 2-phase voltages and currents

A 2-phase voltage system is defined by two voltages in quadrature:

if it is loaded onto a symmetrical impedance it leads to a balanced 2-phase current system:

There is no pulsating power and the instantaneous power is constant:

Figure 1.3.2-phase currents and voltages

The complex representation can also be introduced:

with the expressions of the active and reactive powers:

1.2.3. Balanced 3-phase sinusoidal systems

1.2.3.1. Time expressions

A balanced 3-phase voltage system is composed of three voltages with the same frequency, with the same amplitude and phase shifted by a third of a period with respect to the others. It is thus written as a time expression:

If this voltage system is connected to a symmetrical load (with a circulating impedance matrix), it leads to a balanced current system (Figures 1.4 and 1.5):

with the vector representation shown in Figure 1.4.

Figure 1.4.3-phase currents and voltages

A zero pulsating power is then obtained and the instantaneous power is constant and equal to the active power:

1.2.3.2. Associated complex notations

Complex vectors are associated with balanced voltage and current systems:

[1.4]

If the 3-phase voltage system is applied to a load characterized by a circulating impedance matrix (Figure 1.5) such as:

the expression leads to the phase equation:

in which:

[1.5]

is the impedance of the load.

Figure 1.5.Balanced 3-phase load

This happens as if each of the three phases was loaded with a impedance decoupled from the other two (it is in fact a diagonalization of the impedance matrix that has as an eigenvalue). In those conditions balanced 3-phase systems can be dealt with as independent and decoupled single-phase systems.

1.2.4. Unbalanced 3-phase sinusoidal systems: Fortescue symmetrical components

Voltage and current systems may be unbalanced (different amplitudes depending on the phases or phase-shifts different from 2π/3). Expressions [1.4] and [1.5] are no longer valid and 3-phase equations cannot be replaced by single-phase equations. Generally, the analysis of these systems is very difficult.

However there is a system class, fortunately quite commonplace in electrical engineering, for which there is a mathematical simplification. They are the devices described by a circulating impedance matrix and to which dissymmetrical external conditions are imposed. It can be demonstrated that matrix has three eigenvectors:

respectively associated with the three eigenvalues:

[1.6]

[1.7]

[1.8]

The three above-written impedances are respectively called zero phase-sequence impedance, forward impedance and backward impedance. A transformation matrix can be built from the three eigenvectors:

[1.9]

called “Fortescue's matrix”. Its backward matrix is:

[1.10]

If three variables composing an unbalanced 3-phase system are named Ga, Gb and Gc (voltages, currents, flux, etc.), then the homologous variables G0, Gd and Gi can be defined by:

[1.11]

with, of course, the opposite transition expression:

[1.12]

This last expression shows that if only the forward (positive phase-sequence) part exists, the above-mentioned balanced 3-phase system (Figure 1.4) will be found:

only if the backward (negative phase-sequence) part is not zero:

A balanced 3-phase system is obtained, also known as “backward (negative phase-sequence)”, for which components b and c exchange their roles.

Finally, if only is different from zero:

This is an expression defining a zero phase-sequence 3-phase system in which the three components are identical.

Figure 1.6.Systems: a) forward (positive phase-sequence); b) backward (negative phase-sequence); c) zero phase-sequence

This approach therefore consists of replacing an unbalanced 3-phase system with the superposition of three different balanced 3-phase systems of different natures: forward, backward and zero phase-sequence, which can be studied separately and easily.

Matrix equation:

in which and represent voltage and current unbalanced systems, can be divided into:

and are respectively the impedances of the device in zero phase-sequence, forward and backward modes.

This method, called “Fortescue's symmetric components”, is very convenient for studying and calculating unbalanced sinusoidal 3-phase systems. It is also noticeable that it can be used to study non-sinusoidal balanced 3-phase systems. Indeed it can be demonstrated that 3k rank harmonics create zero phase-sequence systems, that 3k + 1 rank harmonics create forward systems and that 3k - 1 rank harmonics create backward systems.

1.3. Electromagnetism

1.3.1. Primary laws

1.3.1.1. Maxwell's equations

Considering the industrial frequencies used in power systems, displacement currents are neglected, and Maxwell's equations can be written as follows:

[1.13]

[1.14]

[1.15]

[1.16]

[1.17]

1.3.1.2. Ampere's theorem

Equation [1.14] leads to:

[1.18]

The magnetic field circulation on a closed circuit (C) is equal to the algebraic sum of the embraced currents.

1.3.1.3. Faraday's law

Let's consider circuit (C) in Figure 1.7, first assumed to be fixed. Equation [1.13] leads to:

[1.19]

ϕ is the magnetic flux through circuit (C)'s surface S and e is the induced electromotive force (emf) on (C)'s terminals. This is a transformation emf.

If (C) is moving at speed , equation [1.17] gives:

[1.20]

The induced emf at (C)'s terminals is the sum of the transformation emf eT and of emf eV, also called speed emf or emf due to the cut-flux.

Figure 1.7.Production of an emf at the circuit's terminals

1.3.2. Materials and magnetic circuits

When a material is subjected to a magnetic field H, each dV element gains a magnetic moment able to oppose or add itself to H. Those magnetic moments can be considerable for ferromagnetic materials. Magnetic flux density within the material is written:

μ0 is the flux density which would have been created into free space, and [A/m] is the magnetization. This is noted:

The magnetic susceptibility χ usually varies in a very complex way with the field and leads to a B(H) expression presenting a hysteresis (Figure 1.8). Figure 1.8 shows the remanence flux density Br, the coercive field Hc and the initial magnetization characteristics.

According to the hysteresis cycle, “soft” materials can be distinguished from “hard” materials. Soft materials (electrical steel, solid steel, etc.) are characterized by a narrow cycle. Br and Hc are weak. Hc is around 50 to 70 A/m, whereas Br is below 0.1 T. Hard materials (permanent magnets) have a wide cycle. The coercive field Hc is held between 200 and 1,000 kA/m while Br is held between 0.3 T and 1.2 T.

Figure 1.8.Hysteresis cycle

1.3.2.1. Soft ferromagnetic materials

As the hysteresis cycle is narrow, only the initial magnetization curve is taken into consideration. If the material is characterized by a constant χ

1.3.2.1.1. Saturation

The magnetic permeability of ferromagnetic materials depends on the applied field:

Figure 1.9 shows the initial magnetization curve as well as the relative permeability in terms of the magnetic field of a steel frequently used in electrical machines.

Figure 1.9.Initial magnetization curve and variation of the relative permeability of FeV 400-50 HA steel in terms of the field

1.3.2.1.2. Iron losses

Hysteresis losses

When the field in a ferromagnetic material varies with time, losses (called hysteresis losses) appear; they are proportional to the area enclosed by this hysteresis loop. They correspond to the energy required for the orientation of the magnetic moments. Those losses are proportional to frequency f of the excitation currents, as well as to volume V of the magnetic circuit:

Eddy current losses

Let us consider the solid ferromagnetic circuit drawn in Figure 1.10.

Figure 1.10.Flux and induced currents in a solid circuit

When mmf (i.e. magnetomotive force) NI is variable, currents are induced in the conductive iron. The corresponding losses are called “eddy current losses”. In order to reduce those losses the solid material is usually replaced by thin metal sheets insulated from one another. Those losses are given by:

Constant KF depends on the material, e is the metal sheet thickness (about 0.5 mm for electrical machines).

Iron losses

The sum of the hysteresis losses and the eddy current losses are usually gathered under the name “iron losses” (Pfer):

1.3.2.1.3. Magnetic circuits

We have seen that ferromagnetic materials are characterized by an important permeability which enable the magnetic flux to be canalized.

Hopkinson's law

Let us consider the circuit characterized by the average closed path (C) of length l (Figure 1.11). Assuming that field and are colinear, and assuming that H is constant, Ampere's theorem leads to:

Hopkinson's law is obtained:

[1.21]

ε is the magnetomotive force, expressed in [At]. ℜ is the magnetic circuit reluctance, and ℘ its permeance, with:

[1.22]

Figure 1.11.Magnetic field and average flux path

Analogy between a magnetic circuit and an electrical circuit

Hopkinson's law can be represented by Figure 1.12. The flux ϕ circulates within reluctance ℜ of the magnetic circuit, like current I which circulates within resistance R of the electrical circuit. In comparison to Ohm's law, it is therefore noted that mmf ε, flux ϕ and reluctance ℜ are respectively similar to voltage V, current I and electrical resistance R. However there is no equivalence to the notion of electrical losses associated with a resistance. Moreover at constant temperature, R is constant whereas in the presence of saturation, ℜ varies with flux density B.

Figure 1.12.Analogy between a magnetic circuit (a) and an electrical circuit (b)

Series and parallel magnetic circuits

As for electrical circuits:

–the equivalent reluctance ℜeq of several reluctances ℜi connected in series (crossed by the same flux) is:

[1.23]

– the equivalent reluctance ℜeq of several reluctances ℜi connected in parallel (submitted to the same mmf) is:

[1.24]

1.3.2.2. Permanent magnets

1.3.2.2.1. Classification

These are “hard” materials characterized by large hysteresis loops. They operate in plane Ba > 0 and Ha < 0 (Figure 1.13). There are:

– ferrites or ceramics. They are iron oxide-based materials characterized by:

– Alnico or metal magnets also called “Ticonal” and mainly constituted of iron, cobalt, nickel, aluminum and copper, with:

– rare earth magnets, samarium-cobalt (Sm2 Co17, Sm Co5, etc.) and neodymium-iron-boron magnets (NeFeBo). SmCo magnets are characterized by:

NeFeBo magnets are quite sensitive to temperature. In order to avoid their demagnetization they are generally used under 120°C. They are characterized by:

Note that the characteristics of the SmCo, NeFeBo magnets and of the ceramics are quite linear (Figure 1.13). Those magnets are called “rigid magnets”. In this quadrant, this characteristic can be written:

Figure 1.13.Magnetic characteristics of different permanent magnets

1.3.2.2.2. Static behavior

Let us consider the magnetic circuit constituted of a rigid magnet inserted into iron (soft ferromagnetic material of very high permeability) with air-gap e (Figure 1.14a). According to the high permeability of iron, the magnetic field can be neglected in it. If the flux leakage is also overlooked, flux density and magnetic field within the magnet are linked by the external characteristic or load curve:

[1.25]

Sa and La are respectively the cross-section and the length of the magnet.℘e is the air-gap permeance given by:

The P operating point (Figure 1.14b) corresponds to the crossing of this load curve with the magnetic characteristic of the magnet. Note that, within the magnet, the field is negative and the useful portion of the cycle is defined by B > 0 and H < 0.

Figure 1.14.a) Magnetic circuit excited by a magnet; b) operating point P

1.3.2.2.3. Equivalent circuit

Using the Amperian current model of the magnet and if we neglect the fictitious currents inside the magnet and the flux leakages in the magnet, the equivalent circuit of a rigid magnet consists of a magnetomotive force ε in series with magnet reluctance ℜa (Figure 1.15). With:

Figure 1.15.Equivalent circuit of a magnet

For example, the circuit presented in Figure 1.14a can be replaced by the equivalent circuit given by Figure 1.16.

Figure 1.16.Equivalent circuit of a magnetic circuit excited by a magnet

Re and Rf are respectively the reluctances of the air-gap and iron. The magnetic flux Φ is then given by:

1.3.3. Inductances

Electrical circuits are supposed to be in free space.

1.3.3.1. Mutual inductances

Let's consider two circuits (C1) and (C2) having respectively N1 and N2 turns (Figure 1.17). Only (C1) is supposed to be supplied by current I1.

Figure 1.17.Electrical circuits (C1) and (C2)

The total flux in (C2) created by I1 is given by:

M12 is the mutual inductance between (C1) and (C2). With:

[1.26]

M12 is therefore proportional to the product N1N2. In the same way, if only (C2) was supplied by I2, the total flux in (C1) would be:

with:

1.3.3.2. Self-inductances

If the circuit consists of N1 turns, the total flux is:

The self-inductance is then obtained by:

[1.27]

Flux ϕ1 being proportional to N1, it can be noticed that L1 is proportional to (N1)2.

1.3.3.3. Coupled circuits

Let us consider two circuits (C1) and (C2) from Figure 1.17 again, and let us suppose that they are respectively supplied by currents I1 and I2. The total flux created by I1 equals L1I1. A part of this flux, equal to M12I1, crosses circuit (C2). The total flux crossed by (C1) and (C2) is therefore given by:

It is then written in a matricial form:

The co-energy (numerically equal to the energy in a linear case) associated with those two circuits is given by:

and thus:

This result can be generalized to n coupled circuits:

[1.28]

1.3.3.4. Inductances, reluctances, permeances

A magnetic circuit of reluctance ℜ, crossed by magnetic flux ϕ (Figure 1.18) is considered.

Figure 1.18.Magnetic circuit of reluctance ℜ

– ϕ is the flux embraced by one turn of the coil.

The self-inductance of a coil is therefore bound to reluctance ℜ (or to permeance ℘) of the corresponding magnetic circuit by:

[1.29]

This shows that the self-inductance of a coil is forwardly related to the permeance of its magnetic circuit. This is also the case for the mutual inductances between two coils.

1.3.4. Skin effect or Kelvin effect

Let us consider one homogenous conductor with a rectangular cross-section S characterized by permeability μ and conductivity σ, and travelling by current i (Figure 1.19). When i is DC, current density J is constant in the entire cross-section S. J is not constant for an alternating current.

In this case, the field leads to auto-induction phenomena and there is a current concentration on the surface which is all the more important when the frequency is high. The conductor then has a higher electrical resistance than the resistance obtained with forward current; it is sometimes necessary to subdivide the conductors, particularly when frequency increases.

Skin thickness δ is defined as being the thickness of the layer in which most of the current is concentrated.

[1.30]

As an example, for copper, δ approximately equals 1 cm at 50 Hz, and 2 mm at 1 kHz.

Figure 1.19.Schematic representation of the current repartition in a conductor: a) forward current; b) alternating current

1.3.5. Torque calculation using the virtual work principle

1.3.5.1. Single-phase system

Let us consider a single-phase system (Figure 1.20) converting electrical energy into mechanical energy. Electrical energy W1 provided by the source is the sum of energy Wp